EXAMPLE 5 Find a power model Find a power model for the original data. The table at the right shows the typical wingspans x (in feet) and the typical weights y (in pounds) for several types of birds. Biology Draw a scatter plot of the data pairs (ln x, ln y ). Is a power model a good fit for the original data pairs (x, y) ?

EXAMPLE 5 Find a power model SOLUTION Use a calculator to create a table of data pairs (ln x, ln y). Plot the new points as shown. The points lie close to a line, so a power model should be a good fit for the original data. STEP 1 STEP 2

EXAMPLE 5 Find a power model STEP 3 Find a power model y = ax b by choosing two points on the line, such as (1.227, 0.525) and (2.128, 2.774). Use these points to write an equation of the line. Then solve for y. In y – y = m ( ln x – x ) 1 1 In y = 2.5 In x – Equation when axes are ln x and ln y Substitute. Simplify. In y = In x – Power property of logarithms In y – = 2.5(In x – 2.128)

EXAMPLE 5 Writing Reciprocals Y = e ln x – Y = e e – In x Y = x 2.5 Exponentiate each side using base e. Product of powers property Simplify.

EXAMPLE 6 Use power regression Use a graphing calculator to find a power model for the data in Example 5. Estimate the weight of a bird with a wingspan of 4.5 feet. Biology SOLUTION Enter the original data into a graphing calculator and perform a power regression. The model is y = x Substituting x = 4.5 into the model gives y = (4.5) 3.31 pounds. 2.87

GUIDED PRACTICE for Example 5 and 6 9. The table below shows the atomic number x and the melting point y (in degrees Celsius) for the alkali metals. Find a power model for the data. SOLUTION y = x